# Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style.

For (connected, pointed) topological spaces with trivial fundamental group, there is much already said in the literature and there is a term for such spaces: simply-connected spaces. If I want to search the literature for spaces with fundamental groups such as $\mathbb{Z}$, what terms are relevant?

Moreover, what is the story of spaces with prescribed fundamental groups other than the trivial group? I know this is a classical problem. Even in the simply connected case, there is not still a 'classification scheme' for simply connected smooth 4-manifolds. The Wikipedia page here gives one reason why many studies just consider the simply connected case as "the general case of many problems is already known to be intractable."

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Even more, here it reads: "non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups." So, while non-simply connected spaces are more complicated to classify, what has been done in the simplest cases such as small finite groups? – Mathing being Nov 30 '13 at 13:14

The reason why spaces with trivial fundamental group (i.e. simply connected spaces) appear to be more studied than the others in the literature is principally due to the fact that every sufficiently nice space (here it says: connected, locally path connected and semi-locally simply connected) admits an universal cover, and many problems can be reduced to the study of simply connected spaces this way.

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Thanks. The statement about the universal cover appears on page 64 of Hatcher's Algebraic Topology book. While that is a good reason, is there a buzzword for searching about spaces with given fundamental groups? Or is there some kind of table of examples or an atlas of known cases? – Mathing being Nov 30 '13 at 13:02

From the POV of homotopy theory one reason why simply connected spaces are nice is the following consequence of (relative) Hurewicz theorem.

If $X$ and $Y$ are simply connected and $f\colon X\to Y$ induces isomorphism of all homology groups $f$ is a (weak) homotopy equivalence.

This is true for any spaces if one use $\pi_i$ instead of $H_i$ — but homology groups are much, much easier to compute.

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Thank you. Do you have a suggestion on my comment on the other answer? – Mathing being Nov 30 '13 at 13:07
@Mathingbeing Sometimes it makes sense to study spaces with some condition on $\pi_1$: with abelian $\pi_1$ or, say, with nilpotent $\pi_1$ (e.g. in rational homotopy theory). But I don't think conditions like «have $\pi_1$ equal to $\mathbb Z$» are that useful. (Perhaps, one exception is topology of 3- and 4-manifolds, but I don't know much about this area.) – Grigory M Nov 30 '13 at 13:21
Aha. I think what I was looking for partially was the phrase "non-simply connected spaces," for lack of better terms. I'll search more and look forward to more answers. – Mathing being Nov 30 '13 at 13:31

So, I am answering my own question whose answer I found some time ago.

One appropriate term to use to search the literature about spaces with fundamental group $G$ is $K(G,1)$. These are special cases of the Eilenberg-MacLane spaces. Quoting Wikipedia:

Let $G$ be a group and $n$ a positive integer. A connected topological space $X$ is called an Eilenberg–MacLane space of type K(G, n), if it has $n$-th homotopy group $\pi_n(X)$ isomorphic to $G$ and all other homotopy groups trivial.

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